Two statements are logically equivalent if they have the same truth table inputs and outputs. How do I know if one statement can be inferred from another?
Does that just mean for inputs they share, they also share the same outputs?
Let me give an example of what I'm trying to do:
1.) If the car door is unlocked then Jimmy can enter the car
I understand that to mean P->Q
An equivalent statement to P->Q would be one with the same truth table (like ~P OR Q). They are equivalent because they have the same truth table:
P Q P->Q ~P OR Q
T T T T
T F F F
F T T T
F F T T
But I want to determine if another statement can be inferred from statement #1.
2.) If the car door is locked then Jimmy can not enter the car.
~P -> ~Q
Since ~P->~Q has a different truth table I know it's not equivalent and it can't be infered. But I could have a statement like '(P OR Q) AND (P->R)' and I want to know if (Q OR R) can be inferred from it.
Do I just compare the truth table values for Q and R?